Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold

Abstract

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of mathbb {R}^d. For restrictions to the Euclidean ball in odd dimensions, to the rotation group textrm{SO}(3), and to the Grassmannian manifold mathcal {G}_{2,4}, we compute the kernels’ Fourier coefficients and determine their asymptotics. The L_2-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For textrm{SO}(3), the nonequispaced fast Fourier transform is publicly available, and, for mathcal {G}_{2,4}, the transform is derived here. We also provide numerical experiments for textrm{SO}(3) and mathcal {G}_{2,4}.